Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm roughly familiar with the concept of $L^p$ norms -- what they represent and how they are computed -- though I am far from educated in functional analysis in general. For reasons that are more or less obvious, in many practical applications, the $L^2$ norm is used (ie, optimization, etc). One also sees the $L^1$ and $L^{\infty}$ norms used from time to time.

However, rarely do I see any mention of, say, the $L^3$ norm, or really any $L^p$ norm where $p$ is something other than 1, 2 or $\infty$ (or, in some interpretations, 0).

Are there any algorithms or applications that have exploited the other norms? Does convergence in an $L^p$ sense mean something different for these values of $p$? Is there any value of $p$ that can lead to a property not obtainable by $p=1,2,\infty$?

share|cite|improve this question
Relevant:… – Jesse Madnick Sep 22 '12 at 3:43
The $L_0$-"norm", (a misnomer since it is not a norm in the usual sense), which counts the number of non-zero components of a vector, is a convenient notation to deal with sparsity in compressed sensing. – user17762 Sep 22 '12 at 4:21
up vote 4 down vote accepted

One example not mentioned in the MO thread: the $L^4$ norm appears in the Ginzburg-Landau formula for the energy of superconductor.

More generally: we want nonlinear variational models to describe phenomena which do not obey linear superposition. Minimization of $L^2$ norm leads to linear PDE. Minimization of $L^1$ and $L^\infty $ norms leads to badly degenerate things that are barely PDE at all. The norms $L^4$ and $L^{\rm dimension}$ emerge as attractive alternatives.

share|cite|improve this answer
Along these lines, could seeking alternative norms for minimization be a general strategy for solving other problems? – Emily Sep 24 '12 at 18:15
@EdGorcenski One usually minimizes an energy functional, which is not necessarily a norm (i.e., neither homogeneity nor triangle inequality are required). But energy functionals tend to have bits of $L^p$ norms in them. A Google Scholar search shows that $L^2$ and $L^4$ are the most popular ingredients. – user31373 Sep 24 '12 at 18:51

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.